The present invention relates to an apparatus for obtaining a tomogram of a living body by utilizing the nuclear magnetic resonance phenomenon, and more particularly to an image reconstructing device suited to separate chemical shift components from measured resonance signals and to form a tomogram of each chemical shift component, by a high-speed imaging method in which a resonance signal is measured in a state that the strength of a gradient magnetic field is varied with time. Incidentally, the image reconstruction means that a tomogram is reconstructed from a measured resonance signal.
A first method of forming a tomogram of each chemical shift component in the magnetic resonance imaging (hereinafter referred to as "MRI") is described in, for example, an article entitled "Chemical-Shift Imaging with Large Magnetic Field Inhomogeneity" (Magnetic Resonance in Medicine, Vol. 4, No. 5, 1987, pages 452 to 460). In the first method, resonance signals are measured at some different periods of time in a state that the strength of a gradient magnetic field is kept constant, tomograms of an object are reconstructed from measured resonance signals by the two-dimensional Fourier transformation method, and tomograms corresponding to chemical shift components are separated from the reconstructed tomograms. In more detail, each of the chemical shift components is slightly different in resonance frequency from each other. Thus, when a resonance signal is measured at different periods of time, tomograms corresponding to the chemical shift components are different from each other in phase, that is, in the state of nuclear spin. Further, image processing is carried out so that tomograms corresponding to undesired chemical shift components cancel each other on the basis of the above phase difference, to obtain tomograms each formed of a single chemical shift component.
While, a novel imaging method is described in, for example, an article entitled "A Novel Fast Scanning System" (Proceedings of 5th Society of Magnetic Resonance in Medicine, 1986, pages 156 and 157). In this method, a resonance signal is measured in a state that the strength of a gradient magnetic field is varied with time. As a result, the resonance signal can be continuously measured, and thus a time necessary for imaging can be shortened in a great degree.
A general method of reconstructing a tomogram for a resonance signal which is obtained in the abovementioned manner, that is, the so-called correlation method is discussed on page 781 of the Proceedings of the 6th Society of Magnetic Resonance in Medicine held in 1987.
A second method of forming tomograms of a plurality of chemical shift components at the same time from a resonance signal which is measured only once, is discussed on page 230 of the Proceedings of the 6th Society of Magnetic resonance in Medicine held in 1987. The second method utilizes a fact that when the strength of two gradient magnetic fields in different directions is varied with time to obtain two-dimensional tomograms, the difference in resonance frequency between chemical shift components generates blurred tomograms. Let us pay attention to the atomic nucleus of hydrogen, by way of example. When a tomogram is reconstructed in accordance with the resonance frequency of the atomic nucleus of hydrogen contained in water, a resonance signal due to the atomic nucleus of hydrogen contained in fat is subjected to additional phase rotation, and thus a blurred tomogram is superposed on the tomogram due to water. When a tomogram is reconstructed in accordance with the resonance frequency of the atomic nucleus of hydrogen contained in fat, a tomogram due to water is blurred. That is, when a tomogram is reconstructed in accordance with the resonance frequency of the atomic nucleus of hydrogen contained in one of water and fat, a blurred tomogram due to the other substance is superposed on the tomogram due to one substance. When a blurred tomogram can be neglected, the tomogram due to water and the tomogram due to fat can be obtained from the single resonance signal.
The first method has given no consideration to the matter that tomograms due to chemical shift components may be reconstructed at high speed, by using a time-varying gradient magnetic field. The reason for this is as follows.
Now, let us express the tomogram formed of the i-th chemical shift component by .rho..sub.i (x, y), where x and y designate two-dimensional coordinate values on the tomogram, with the origin at the center of a field of view. Further, let us express the deviation of a resonance frequency of each chemical shift component from a reference frequency by .DELTA..omega..sub.i. It is to be noted that the reference frequency means the resonance frequency corresponding to a specific chemical shift component under a static magnetic field. Then, the resultant resonance signal S(t) of respective resonance signals of chemical shift components is given by the following equation: ##EQU1## where ##EQU2## and j=.sqroot.1. Further, G.sub.x and G.sub.y indicate the strength of x- and y-gradient magnetic fields, respectively, .gamma.: a nuclear gyromagnetic ratio, and C: a constant.
The inhomogeneity of static magnetic field and the T.sub.2 effect (namely, transverse relaxation time effect) have no connection with the following explanation, and hence explanation thereof will be omitted. Now, let us consider a case where G.sub.x and G.sub.Y are constant as in the two-dimensional Fourier transformation method, for example, a case where G.sub.x (t)=G.sub.x.sup.o (namely, a constant) and G.sub.y (t)=0. The resonance signal S(t) is rewritten as follows: ##EQU3## Thus, tomograms formed of different chemical shift components are shifted from each other by an amount equal to .DELTA..omega..sub.i /(YG.sub.x.sup.o). As a result, an overlapping tomogram is formed. In a case where G.sub.x and G.sub.y vary with time, however, signal components due to different chemical shift components do not appear as signal components which are merely shifted from each other, but interact on each other in a complicated manner. Thus, it is impossible to reconstruct tomograms corresponding to chemical shift components at the same time. That is, the first chemical shift imaging method is inapplicable to a case where both G.sub.x and G.sub.y vary with time.
While, according to the second method, the point spread function (hereinafter referred to as "PSF") at a focused tomogram is equal in average power (that is, average brightness) to the PSF at a tomogram which is out of focus, and hence it is necessary to remove the influence of the tomogram which is out of focus, on the focused image. In other words, the shape of the PSF is a function for indicating the degree of blur, and a tomogram due to a desired chemical shift component is equal in power to a tomogram due to an undesired chemical shift component. Hence, it is impossible to obtain focused tomograms of a plurality of chemical shift components at the same time.
Various methods of reconstructing tomograms from the resonance signal of the equation (1) have been known. The most general one of the above methods is the correlation method. Accordingly, the correlation method which is described in IEEE Transactions on Medical Imaging, Vol. 7, No. 1, pp. 26-31, March 1988, for example, will be used in the following explanation. It is needless to say that various methods other than the correlation method can be used for reconstructing tomograms.
Now, let us express a tomogram which is reconstructed in accordance with the k-th chemical shift component, by .rho..sub.k (x, y). Then, .rho..sub.k (x, y) is generally given by a complex function as follows: EQU .rho..sub.k (x, y)=.intg.dts(t)exp[+j.gamma.{kx(t)x+ky(t)y}.multidot.exp[-j.DELTA..omega.. sub.k t].multidot.W(t) (3)
where W(t) indicates a weight coefficient for controlling a PSF.
When the equation (1) is substituted in the equation (3), we can obtain the following equation: ##EQU4## where a mathematical sign * indicates two-dimensional convolution which is given by the following equation: ##EQU5## Further, in the equation (4), h.sub.ki (x, y) indicates a PSF which is given by the following equation: EQU h.sub.ki (x, y)=C.intg.dt exp[j.gamma.{kx(t)x+ky(t)y].multidot.exp[+j(.DELTA..omega..sub.i -.DELTA..omega..sub.k)t].multidot.W(t) (6)
In more detail, h.sub.kk indicates a focused PSF, and h.sub.ki (k.noteq.i) a PSF which is out of focus. The PSF h.sub.ki can be obtained from the time-variation pattern of each gradient magnetic field and the weight coefficient W(t), and hence is considered to be a known function.
From the equation (6), we can obtain the following equation: EQU .intg.dxdy.vertline.h.sub.kk (x, y).vertline..sup.2 =.intg.dxdy.vertline.h.sub.ki (x, y).vertline..sup.2 for all i (7)
As is evident from the equation (7), it is impossible to neglect the influence of chemical shift components other than the k-th chemical shift component on the tomogram .rho..sub.k given by the equation (4). This is true of a case where only two chemical shift components exist as in the atomic nucleus of hydrogen.